# Quiz Chapter 08: Gauss-Seidel Method

 MULTIPLE CHOICE TEST GAUSS-SEIDEL METHOD

1. A square matrix $\left[ A \right]_{n \times n}$ is diagonally dominant if

2. Using $\begin{bmatrix} x_{1}&x_{2}&x_{3} \\ \end{bmatrix} = \begin{bmatrix} 1&3&5 \\ \end{bmatrix}$ as the initial guess, the value of $\begin{bmatrix} x_{1}&x_{2}&x_{3} \\ \end{bmatrix}$ after three iterations of Gauss-Seidal method is

• $\begin{bmatrix} 12&7&3 \\ 1&5&1 \\ 2&7&-11 \\ \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ \end{bmatrix} = \begin{bmatrix} 2 \\ -5 \\ 6 \\ \end{bmatrix}$

3. To ensure that the following system of equations,

• $2x_{1} + 7x_{2} - 11x_{3} = 6$
• $x_{1} + 2x_{2} + x_{3} = -5$
• $7x_{1} + 5x_{2} + 2x_{3} = 17$

converges using the Gauss Seidal method, one can rewrite the above equations as follows:

4. For $\begin{bmatrix} 12&7&3 \\ 1&5&1 \\ 2&7&-11 \\ \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ \end{bmatrix} = \begin{bmatrix} 22 \\ 7 \\ -2 \\ \end{bmatrix}$ and using $\begin{bmatrix} x_{1}&x_{2}&x_{3} \\ \end{bmatrix} = \begin{bmatrix} 1&2&1 \\ \end{bmatrix}$ as the initial guess, the value of $\begin{bmatrix} x_{1}&x_{2}&x_{3} \\ \end{bmatrix}$ are found at the end of each iteration as

 Iteration # $x_{1}$ $x_{2}$ $x_{3}$ $1$ $0.41666$ $1.1166$ $0.96818$ $2$ $0.93989$ $1.0183$ $1.0007$ $3$ $0.98908$ $1.0020$ $0.99930$ $4$ $0.99898$ $1.0003$ $1.0000$

At what first iteration number would you trust at least $1$ significant digit in your solution?

5. The algorithm for the Gauss-Seidal method to solve $\left[ A \right] \left[ X \right] = \left[ C \right]$ is given as follows when using nmax iterations. The initial value of [X] is stored in [X].

6. Thermistors measure temperature, have a nonlinear output and are valued for a limited range. So when a thermistor is manufactured, the manufacturer supplies a resistance vs. temperature curve. An accurate representation of the curve is generally given by

• $\dfrac{1}{T} = a_{0} + a_{1} \ln \left( R \right) + a_{2} \left[ \ln \left( R \right) \right]^{2} + a_{3} \left[ \ln \left( R \right) \right]^{3}$

Where $T$ is temperature in Kelvin, $R$ is resistance in ohms, and $a_{0},a_{1},a_{2},a_{3}$ are constants of the calibration curve. Given the following for a thermistor

 $R$ (ohms) $T \left( ^{\circ} C \right)$ $1101.0$ $25.113$ $911.3$ $30.131$ $636.0$ $40.120$ $451.1$ $50.128$

the value of the temperature in $^{\circ}C$ for a measured resistance of $900$ ohms most nearly is